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Let x and y be random vectors of the same dimension. The covariance of x+y is:$\DeclareMathOperator{\Cov}{Cov}$ $$\Cov(\mathbf{x}+\mathbf{y})= \Cov(\mathbf{x})+\Cov(\mathbf{y})+ \Cov(\mathbf{x,y})+\Cov(\mathbf{y,x})$$ Is it correct $\;\Cov(\mathbf{x,y})+\Cov(\mathbf{y,x}) =2\Cov(\mathbf{x,y})$?

Bernard
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    What do you mean when you apply covariance to a single argument? Covariance is a measure of the joint variability of two random variables. – Eric Towers Sep 19 '20 at 18:04
  • When you take the variance of all single argument functions in the first line, your conclusion is correct. – Fakemistake Sep 20 '20 at 05:11

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Just use the definition, note that if $X$ and $Y$ be random vector, so $$Cov(X,Y)=\mathbb{E}(XY)-\mathbb{E}(X)\mathbb{E}(Y)$$where $\mathbb{E}(X)$ denote expected value of $X$. Can you continue from here?